3.900 \(\int \frac{x^5}{(1-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^2}{2 \sqrt{1-x^4}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

[Out]

x^2/(2*Sqrt[1 - x^4]) - ArcSin[x^2]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0118562, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {275, 288, 216} \[ \frac{x^2}{2 \sqrt{1-x^4}}-\frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4]) - ArcSin[x^2]/2

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x^5}{\left (1-x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (1-x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{x^2}{2 \sqrt{1-x^4}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,x^2\right )\\ &=\frac{x^2}{2 \sqrt{1-x^4}}-\frac{1}{2} \sin ^{-1}\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0155917, size = 26, normalized size = 0.96 \[ \frac{1}{2} \left (\frac{x^2}{\sqrt{1-x^4}}-\sin ^{-1}\left (x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 - x^4)^(3/2),x]

[Out]

(x^2/Sqrt[1 - x^4] - ArcSin[x^2])/2

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 62, normalized size = 2.3 \begin{align*} -{\frac{\arcsin \left ({x}^{2} \right ) }{2}}-{\frac{1}{4\,{x}^{2}+4}\sqrt{- \left ({x}^{2}+1 \right ) ^{2}+2+2\,{x}^{2}}}-{\frac{1}{4\,{x}^{2}-4}\sqrt{- \left ({x}^{2}-1 \right ) ^{2}+2-2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-x^4+1)^(3/2),x)

[Out]

-1/2*arcsin(x^2)-1/4/(x^2+1)*(-(x^2+1)^2+2+2*x^2)^(1/2)-1/4/(x^2-1)*(-(x^2-1)^2+2-2*x^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.52752, size = 42, normalized size = 1.56 \begin{align*} \frac{x^{2}}{2 \, \sqrt{-x^{4} + 1}} + \frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{4} + 1}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

1/2*x^2/sqrt(-x^4 + 1) + 1/2*arctan(sqrt(-x^4 + 1)/x^2)

________________________________________________________________________________________

Fricas [B]  time = 1.45513, size = 113, normalized size = 4.19 \begin{align*} -\frac{\sqrt{-x^{4} + 1} x^{2} - 2 \,{\left (x^{4} - 1\right )} \arctan \left (\frac{\sqrt{-x^{4} + 1} - 1}{x^{2}}\right )}{2 \,{\left (x^{4} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/2*(sqrt(-x^4 + 1)*x^2 - 2*(x^4 - 1)*arctan((sqrt(-x^4 + 1) - 1)/x^2))/(x^4 - 1)

________________________________________________________________________________________

Sympy [A]  time = 1.44503, size = 46, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{i x^{2}}{2 \sqrt{x^{4} - 1}} + \frac{i \operatorname{acosh}{\left (x^{2} \right )}}{2} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{x^{2}}{2 \sqrt{1 - x^{4}}} - \frac{\operatorname{asin}{\left (x^{2} \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(-x**4+1)**(3/2),x)

[Out]

Piecewise((-I*x**2/(2*sqrt(x**4 - 1)) + I*acosh(x**2)/2, Abs(x**4) > 1), (x**2/(2*sqrt(1 - x**4)) - asin(x**2)
/2, True))

________________________________________________________________________________________

Giac [A]  time = 1.15225, size = 38, normalized size = 1.41 \begin{align*} -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} - \frac{1}{2} \, \arcsin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-x^4 + 1)*x^2/(x^4 - 1) - 1/2*arcsin(x^2)